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Physical Review Letters

Semiclassical Analysis of Quantum Spectral Functions for a Chaotic Billiard

B. Mehlig, D. Boosé, Kirsten Müller

Published Jul 3, 1995• 11 citations
Publisher

Abstract

For a chaotic billiard, we calculate the quantum spectral functions associated with the position operator $\stackrel{^}{X}$ and the momentum operator ${\stackrel{^}{P}}_{x}$. Semiclassical analysis shows that the quantum spectral function is determined to a large extent by classical ergodic autocorrelations inherent in the Weyl part. For the first time, we calculate periodic orbit corrections to the spectral function.

DOI
10.1103/physrevlett.75.57

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For a chaotic billiard, we calculate the quantum spectral functions associated with the position operator \stackrel{^}{X} and the momentum operator {\stackrel{^}{P}}_{x}. Semiclassical analysis shows that the quantum spectral function is determined to a large extent by classical ergodic autocorrelations inherent in the Weyl part. For the first time, we calculate periodic orbit corrections to the spectral function.
Using a recently defined quantum spectral function and the method of closed-orbit theory we studied the spectra of the half-circular and the quarter circular billiard systems for arbitrary two points.The quantum spectra contain rich information of all classical orbits connecting two arbitrary points in well.Our calculation demonstrates that the peak positions in the Fourier-transformed quantum spectra match accurately with the lengths of the classical orbits.The two examples show evidently that semi-classical method provides a bridge between quantum and classical mechanics.
The use of wave packet to analyze the dynamics of quantum mechanical systems is an increasingly important method to the study of the classical-quantum correspondence.Using the quantum Gaussian wave packet analysis method,we calculate the autocorrelation function of the rectangular billiard,the peak positions of the autocorrelation function match well with the periods of the classical periodic orbits,which show that the period of the classical orbits can be produced by the time-dependent quantum wave packet method.We also discuss wave packet revivals and fractional revivals in the rectangular billiard,the results show that there are exact revival for all wave packet at each revival time.We find additional cases of exact revivals with short revival times for zero-momentum wave packets initially located at special symmetry point inside the billiard.
We establish a duality between the quantum wave vector spectrum and the eigenmodes of the classical Liouvillian dynamics for integrable billiards. Signatures of the classical eigenmodes appear as peaks in the correlation function of the quantum wave vector spectrum. A semiclassical derivation and numerical calculations are presented in support of the results. These classical eigenmodes can be observed in physical experiments through the autocorrelation of the transmission coefficient of waves in quantum billiards. Exact classical trace formulas of the resolvent are derived for the rectangle, equilateral triangle, and circle billiards. We also establish a correspondence between the classical periodic orbit length spectrum and the quantum spectrum for integrable polygonal billiards.
Quantum billiards has attracted much interest recent years.There are many works on the two-dimensional billiards(2D).Compare with the 2D billiards,the three-dimensional(3D) billiards approaches with the actual system better.Taking the cubic billiards as an example,using the closed orbit theory(COT),we find the classical open orbits and study the correspondence between the quantum spectra and the length of classical open orbits in the 3D billiard system.The peaks' locations of the quantum spectra match with the length of the classical open orbits very well.This will help us to analyze the transport property through an open billiards.By this method,the physical image is clear,the computation is easier and it can help us to understand the character of some chaotic systems better.This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.